27 problems

Mersenne-period conjecture. For all $n\geq 2$,

Let $R$ be a strongly discrete coherent ring. For $n\a0\geq 1$, let $R[X_1,\ldots,X_n]$ be a polynomial ring, and let $\prec$ be a rational monomial order on its monomials. For a f…

Let $A$ be a regular ring of finite Krull dimension, and let $n\ge 1$. A finitely generated projective module over $A[x_1,\ldots,x_n]$ is extended from $A$ if it is obtained from a…

Winning-strategy conjecture. Player A has a winning strategy on $\bar{K}[x_1,\dots,x_n]$ for any $n\geq 1$.

Yan's conjecture. If $d$ is a simple derivation of $R_n$, then $\operatorname{Aut}(R_n)_d$ is conjugate in $\operatorname{Aut}(R_n)$ to a subgroup of the group of translations.

Let $K$ be a field of characteristic zero and let $K[T]=K[t_1,\dots,t_n]$. A derivation $D$ of a ring is locally nilpotent if for every $a$ there is an integer $m(a)$ such that…

Let $T$ denote the polynomial setting used in the paper, and let $\mathbb{N}$ denote the positive integers. A polynomial pattern $\mathcal{F}$ is partition regular over a domain if…

For $d\geq 1$, let $P_d$ be the set of polynomials in countably many variables with nonnegative integer coefficients, constant term, and degree at most $d$ in each variable. A patt…

For $d\geq 1$, let $P_d$ be the set of polynomials in countably many variables with nonnegative integer coefficients, constant term, and degree at most $d$ in each variable. A fini…

Let $K$ be a field, let $n\geq 2$, and let $D$ be a simple derivation of the polynomial ring $K[x_1,\ldots,x_n]$. A $K$-subspace $M$ of a ring is a Mathieu-Zhao space if, whenever…

Let $F$ and $G$ be algebraically independent polynomials in $\mathbb{C}[x_1,\ldots,x_n]$. Assume that each generates its own centralizer in $\mathbb{C}[x_1,\ldots,x_n]$, that…

Let $K$ be the algebraically closed field of characteristic zero fixed in the paper, and let $\delta=I-\phi$ be an E-derivation of $K[x_1,x_2,x_3]$, where, for $i=1,2$,…

Let $\mathcal{S}$ be the polynomial ring in $n+1$ variables, and let $f_1,\ldots,f_s$ be general forms of degrees $d_1,\ldots,d_s$. Set … For a power series $P=\sum_{i\geq0}a_it^i$…

Let $\mathcal{S}$ be the polynomial ring in $n+1$ variables, and let $f_1,\ldots,f_s$ be general forms of degrees $d_1,\ldots,d_s$. Set … For a power series $P=\sum_{i\geq0}a_it^i$…

Let $K[x_1,x_2,\ldots,x_n]$ be a polynomial ring, and let $D$ be a Shamsuddin derivation, meaning a derivation of the form … where $a_i,b_{i,j}\in K[x]$ and $a_i\neq a_l$ for…

The two-dimensional Centralizer Conjecture over $D$. Suppose

Let $k$ be a field. Define $s$ by $|k|=\aleph_s$ if $k$ is infinite, and let $s=0$ otherwise. For a non-negative integer $n$, set $R=k[X_1,\ldots,X_n]$ and let $Q(R)$ denote its fr…

Murthy's complete intersection conjecture. For every ideal $I$ in $A$,

Let $A=\mathbb{Z}[T_1,\dots,T_n]$, and let $A^+=\mathbb{N}[T_1,\dots,T_n]$. Let $B$ be a subring of $A$ and $I$ an ideal of $B$. Write $QF(B)$ for the quotient field of $B$. Kala's…

Let $k$ be a field of characteristic zero, and let … be an epimorphism of $k$-algebras. Set $I=\ker(\varphi)$, and let $\mu(I)$ denote the minimal number of generators of $I$. Epi-…

Let $k$ be an algebraically closed field of characteristic zero and let $R=k[x_1,\ldots,x_m]$. Let $\mathfrak q$ be a nonzero ideal and let $\mathfrak a_\bullet$ be a graded sequen…

Let $R$ be a polynomial ring over a field, and let $I\subseteq R$ be an ideal with minimal homogeneous generators whose number and degrees are fixed. The regularity-bound conjectur…

Let $K$ be a field and let $R=K[x_1,\ldots,x_N]$ be a polynomial ring. For fixed positive integers $n$ and $d_1,\ldots,d_n$, let $I\subseteq R$ be an ideal generated by $n$ homogen…

Let $A$ be a commutative ring with one, let $A^{[n]}$ be a polynomial ring over $A$, let $a,b\in A^{[n]}$, and let $g\in A^{[n]}[y]$. Coordinate equivalence conjecture. If $A$ is a…

Let $A$ be a commutative ring with one, let $A^{[n]}$ denote a polynomial ring in $n$ variables over $A$, and let $f\in A^{[n]}$. A polynomial $f$ is a stable coordinate if it is a…